ISSN 0032-9460, Problems of Information Transmission, 2007, Vol. 43, No. 1, pp. 57–68.
Pleiades Publishing, Inc., 2007.
Original Russian Text
B.D. Kudryashov, K.V. Yurkov, 2007, published in Problemy Peredachi Informatsii, 2007, Vol. 43, No. 1, pp. 67–79.
Random Coding Bound for the Second Moment
of Multidimensional Lattices
B. D. Kudryashov and K. V. Yurkov
St. Petersburg State University of Aerospace Instrumentation
Received October 13, 2006; in ﬁnal form, December 14, 2006
Abstract—Eﬃciency of lattice quantization depends on the parameter of a lattice called the
normalized second moment of the Voronoi polyhedron. We apply random-coding methods to
study lattices generated by q-ary linear codes. We prove that in this class there are lattices
with the normalized second moment close to the theoretically attainable limit.
It is well known that, for an arbitrary continuous stationary source, the theoretical limit of
encoding eﬃciency is its rate-distortion function, or ε-entropy . A discrete-time encoder of a
continuous source is often referred to as a vector quantizer, and the code as a codebook . Clas-
sical results on the existence of asymptotically optimal codebooks are proved by random coding
methods without any restrictions on the codebook structure. It is clear that the encoding complex-
ity for codebooks of an arbitrary structure is proportional to the codebook size; i.e., the encoding
complexity for sequences of length n with rate r bits per sample is proportional to 2
A large step towards reducing the quantization complexity is due to the use of quantizers based
on multidimensional lattices . In particular, Conway and Sloane proposed a simple encoding
algorithm, which, in essence, consists in a linear (in n) number of scalar quantization operations
and subsequent exhaustive search. The complexity of this quantization grows exponentially in the
dimension n of the vectors; however, it is determined by the number of points in the fundamental
region of the lattice and does not depend on the encoding rate r.
Basic results on the analysis of the lattice quantization eﬃciency are due to Zador , who
has shown that, in the case of small quantization errors, characteristics of the quantizer are deter-
mined by the shape of the Voronoi polyhedra of the lattice points; more precisely, by their so-called
normalized second moment.
Several constructions of known lattices with good values of the second moment are presented in
the monograph . The fact that constructions of such lattices are closely related to constructions
of block error-correcting codes (Hamming, Golay, Nordstrom–Robinson, etc.) suggests the possi-
bility of searching for quantizers with a good relation between the complexity and error among
quantizers based on block codes. In particular, in  it is shown that there exist asymptotically
optimal lattices over q-ary linear codes as q tends to inﬁnity.
In the present paper we analyze multidimensional lattices over linear codes with an arbitrary
alphabet size q. Of the most practical interest are lattices over binary codes since many good
binary codes are known. Moreover, quantization for such lattices reduces to decoding in soft-
decision channels, which would allow one to use numerous eﬃcient decoding procedures for linear
codes. The main result of the paper is computing the asymptotic lower bound on the quantization